I am able to calculate (estimate) the acceleration around a track (path of unequally spaced points in two dimensions) for a given speed profile. However I now wish to find the speed at each point that will give me constant acceleration, or an acceleration close to, but not exceeding, a maximum. My maths skills start to fail me here for two reasons. Firstly the acceleration is both a function of speed and rate of change of speed. So you cannot solve for a point as the solution is also a function of the solution at other points. The second issue is that you may need to start to "slow down" for a corner more than one point before. I think I can fix the second problem if I could do the first.
If you want some formulas I calculated acceleration to be...
$$ Amax = \sqrt{{\left(\frac{dS}{dt}\right)_n}^2 + ({Ac_n.S_n^2})^2} $$
As Amax is my target acceleration, and constant, i get...
$$ Amax^2 = {\left(\frac{dS}{dt}\right)}_n^2 + ({Ac_n.S_n^2})^2 $$
I also have approximations I'm happy with for...
$$ \left(\frac{dS}{dt}\right)_n = f(S_n,S_{n-1}) $$
$ Ac_n $ is known
Where Ac is acceleration at constant unit speed. I'm happy estimating acceleration due to change in speed for a known set of speeds at each point but no idea how to get these.
I think a numerical method would be easiest to implement. Iterating until a satisfactory solution is achieved. If there is a method to "solve" the problem i would consider this but I think it would be beyond the computational tools I have to address this problem.